i think the way i would do this is make your own linear actuator from a modified r/c servo.

mod the servo so it runs free (eliminate the pot's effectiveness) and have it drive a chain. this chain turns the nuts on each long machine screw... the nuts will spin, but wont move laterally so the whole bolt moves in and out on each side. this gives you a nice controlled linear motion, you can design it to work underneath your screen, just attach a screen tray to the long bolts...

crude drawing, but this is the idea-

or, put one motor on each screw. stronger, and you could then have a horizontal tilt adjustment...